If some elements of \(\mathbf {Y}\)  are continuous and others discrete, then
\(f_Y\)  is often called a “mixed density”. In this case, an extended likelihood can be
constructed from the distribution of \(\mathbf {Y}\)  joint with the elements of \(\mathbf {U}\) 
that correspond only to the discrete variables; see [1]}.
Similarly,
if some individual elements \(Y_j\)  have distributions that are mixtures
of continuous and discrete distributions (such as a zero-inflated
continuous distribution) then an extended likelihood can also be constructed
for this case; see [2]} for how to do so.
